In [1]}, Burgeet et al. discussion the M-R-number \(m_j(G_1, G_2)\) , where both \(G_1\)  and \(G_2\)  is a complete multipartite graph. Recently the M-R-number \(m_j(G_1, G_2)\)  have been study for special classes, see [2]}, [3]}, [4]}, [5]} and its references. Which can be naturally extended to several colors, see [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}. In [10]}, Lusiani et al. determined the M-R-number \(m_j(K_{1,m},G)\) , for \(j = 2,3\)  where \(G\)  is a \(P_n\)  or a \(C_n\) .
